How to implement a moment ODE model

Moment ODEs are powerful models based on ordinary differential equations that allow to consider stochastic dynamics. This can be both intrinsics stochasticity, see in

• Lee C.H. et al., A moment closure method for stochastic reaction networks. The Journal of Chemical Physics, 130(13), 134107, 2013,

or mixtures of intrinsic and extrinsic stochasticity, see in

• Zechner, C. et al., Moment-based inference predicts bimodality in transient gene expression. PNAS, 109(21), 8340-8345, 2012.

To calibrate such models, for instance FACS data can be used.

The mathematical formulation of those model consists of ODEs for the mean of each species, of the covariance components between each species and higher order terms if desired. It can be encoded in the Data 2 Dynamics software as follows:

STATES
mu_1        C   "nmol/l"      "conc."
mu_2        C   "nmol/l"      "conc."
...
cov_1_1     C   "nmol/l^2"    "conc.^2"
cov_1_2     C   "nmol/l^2"    "conc.^2"
cov_2_2     C   "nmol/l^2"    "conc.^2"
...

Please note that component cov_1_2 and cov_2_1 will be identical.

ODES
"mathematical expression for mu_1"
"mathematical expression for mu_2"
...
"mathematical expression for cov_1_1"
"mathematical expression for cov_1_2"
"mathematical expression for cov_2_2"
...

The OBSERABLES and ERRORS section, for instance for FACS measurements of mu_1 and mu_2, can be defined as follows

OBSERVABLES
obs_mu_1    C   "au"  "conc."   1   1   "scale_obs_mu_1 * (mu_1)"
obs_mu_2    C   "au"  "conc."   1   1   "scale_obs_mu_2 * (mu_2)"
...

ERRORS
obs_mu_1    "sqrt(cov_1_1 + sd_mu_1)"
obs_mu_2    "sqrt(cov_2_2 + sd_mu_2)"

The error model parameters sd_mu_1 and sd_mu_2 implement additional measurement noise. Please note that these OBERSERVABLES do not take into account correlation information about mu_1 and m_2 that might be available. If such information should be considered a more complicated likelihood function has to be implemented manually.